Analysis of parameterized quadratic eigenvalue problems in computational acoustics with homotopic deviation theory

This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partl...

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Bibliographic Details
Published in:Numerical linear algebra with applications Vol. 13; no. 6; pp. 487 - 512
Main Authors: Chaitin-Chatelin, F., van Gijzen, M. B.
Format: Journal Article
Language:English
Published: Chichester, UK John Wiley & Sons, Ltd 01.08.2006
Subjects:
acoustic wave equation
critical point
homotopic deviation
impeding boundary
kernel point
quadratic eigenproblem
ISSN:1070-5325, 1099-1506
Online Access:Get full text
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Summary:This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partly impedance, with a complex impedance parameter ζ. The admittance t = 1/ζ is the classical homotopy parameter. In particular, we study the spectrum when t → ∞. We show that in the limit part of the eigenvalues remain bounded and converge to the so‐called kernel points. We also show that there exist the so‐called critical points that correspond to frequencies for which no finite value of the admittance can cause a resonance. Finally, the physical interpretation that the impedance condition is transformed into a pressure release condition when |t| → ∞ enables us to give the kernel points in closed form as eigenvalues of the discrete Dirichlet problem. Copyright © 2006 John Wiley & Sons, Ltd.
Bibliography:ArticleID:NLA484
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ISSN:1070-5325
1099-1506
DOI:10.1002/nla.484